Hexaoctagonal tiling

hexaoctagonal tiling
Hexaoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (6.8)2
Schläfli symbol r{8,6} or
Wythoff symbol 2 | 8 6
Coxeter diagram
Symmetry group [8,6], (*862)
Dual Order-8-6 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.

Constructions

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,6,1+], gives [(8,8,3)], (*883). Removing the mirror between the order 2 and 8 points, [1+,8,6], gives [(4,6,6)], (*664). Removing two mirrors as [8,1+,6,1+], leaves remaining mirrors (*4343).

Four uniform constructions of 6.8.6.8
Uniform
Coloring
Symmetry [8,6]
(*862)
 [(8,3,8)] = [8,6,1+]
(*883)
 [(6,4,6)] = [1+,8,6]
(*664)
 [1+,8,6,1+]
(*4343)
Symbol r{8,6} r{(8,3,8)} r{(6,4,6)}
Coxeter
diagram 		= = =

Symmetry

The dual tiling has face configuration V6.8.6.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4343), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*43) orbifold. These are subsymmetries of [8,6].


[1+,8,4,1+], (*4343)
[(8,4,2+)], (2*43)
Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
{8,6} t{8,6}
 r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
V86 V6.16.16 V(6.8)2 V8.12.12 V68 V4.6.4.8 V4.12.16
Alternations
[1+,8,6]
(*466) 		[8+,6]
(8*3) 		[8,1+,6]
(*4232) 		[8,6+]
(6*4) 		[8,6,1+]
(*883) 		[(8,6,2+)]
(2*43) 		[8,6]+
(862)
h{8,6} s{8,6} hr{8,6} s{6,8} h{6,8} hrr{8,6} sr{8,6}
Alternation duals
V(4.6)6 V3.3.8.3.8.3 V(3.4.4.4)2 V3.4.3.4.3.6 V(3.8)8 V3.45 V3.3.6.3.8
Symmetry mutation of quasiregular tilings: (6.n)2
Symmetry
*6n2
[n,6] 		Euclidean Compact hyperbolic Paracompact Noncompact
*632
[3,6] 		*642
[4,6] 		*652
[5,6] 		*662
[6,6] 		*762
[7,6] 		*862
[8,6]... 		*∞62
[∞,6] 		 
[iπ/λ,6]
Quasiregular
figures
configuration
6.3.6.3
6.4.6.4
6.5.6.5
6.6.6.6
6.7.6.7
6.8.6.8
6.∞.6.∞
6.∞.6.∞
Dual figures
Rhombic
figures
configuration
V6.3.6.3
V6.4.6.4
V6.5.6.5
V6.6.6.6
V6.7.6.7
V6.8.6.8
V6.∞.6.∞
Dimensional family of quasiregular polyhedra and tilings: (8.n)2
Symmetry
*8n2
[n,8] 		Hyperbolic... Paracompact Noncompact
*832
[3,8] 		*842
[4,8] 		*852
[5,8] 		*862
[6,8] 		*872
[7,8] 		*882
[8,8]... 		*∞82
[∞,8] 		 
[iπ/λ,8]
Coxeter
Quasiregular
figures
configuration
3.8.3.8
4.8.4.8
8.5.8.5
8.6.8.6
8.7.8.7
8.8.8.8
8.∞.8.∞ 		 
8.∞.8.∞

See also

Wikimedia Commons has media related to Uniform tiling 6-8-6-8.

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.